10 papers
A note on bounded ratios
Lorenzo Baldi, Mario Kummer
We prove that the set of bounded ratios $\BR(X)$ on a semialgebraic set is the convex cone of linear forms that are nonnegative on the tropicalization $\trop(X)…
Lorentzian polynomials and matroids over triangular hyperfields 2: Analytic aspects
Matthew Baker, June Huh, Mario Kummer +1
Brändén and Huh showed that Lorentzian polynomials unify Hodge-Riemann relations in combinatorics: their supports are M-convex, and every M-convex set supports a Lorentzian polynom…
Ulrich sheaves and determinantal representations for higher secant varieties of curves
Daniele Agostini, Mario Kummer, Jinhyung Park
We show that higher secant varieties of smooth projective curves have symmetric admissible determinantal representations with symmetric Ulrich sheaves of rank one if the embedding…
Totally real divisors on curves
Lorenzo Baldi, Mario Kummer, Daniel Plaumann
Since the works of Krasnov and Scheiderer, there has been an interest in studying effective totally real divisors on a curve X defined over a real closed field, i.e., effective div…
Non-negative polynomials on generalized elliptic curves
Mario Kummer, Aljaž Zalar
We study the cone of non-negative polynomials on generalized elliptic curves. We show that the zero set of every extreme ray has dense real points. If a generalized elliptic curve…
Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects
Matthew Baker, June Huh, Mario Kummer +1
Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space $\mat…
Representation theory for polymatroids
Matthew Baker, June Huh, Donggyu Kim +2
We develop a theory of representations of (discrete) polymatroids over tracts in terms of Plücker coordinates and suitable Plücker relations. As special cases, we recover polymatro…
Two results on the Convex Algebraic Geometry of sets with continuous symmetries
Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes
We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral…
Maximal Mumford Curves from Planar Graphs
Mario Kummer, Bernd Sturmfels, Raluca Vlad
A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves.…
Three results related to the half-plane property of matroids
Mario Kummer, David Sawall
We settle three problems from the literature on stable and real zero polynomials and their connection to matroid theory. We disprove the weak real zero amalgamation conjecture by S…